#34177642

order	20224
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #7845

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₉
commutator	C₂²×C₇₉	C₄³
Fitting	C₇₉×#1744	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₄²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1744	D₇₉
hypocenter	C₇₉	#7845
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ #1744

486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 3 generators on 115 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,81,82,83)(85,90)(86,92)(91,97)(93,98)(101,106)(105,111)(107,113)(112,115);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(84,86)(85,89)(87,93)(88,92)(90,96)(91,95)(94,98)(97,99)(100,102,103,108,104,109,110,114)(101,105,107,112,106,111,113,115);(84,85,87,91,88,90,94,97)(86,89,93,95,92,96,98,99)(100,101)(102,105)(103,107)(104,106)(108,112)(109,111)(110,113)(114,115)

Polycyclic presentation:

2^8,79|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,c^a=cgh,c^b=cg,i^b=i^78

Conjugacy classes & character table

3232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 58941);